Finite Group Databasebeta

← #22561453

Subgroups — #22561453

Order 8576.

298 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524041C211yesmaximal, characteristic
2C2×#14523968C211yesmaximal, characteristic
3C2×#14523967C211yesmaximal, characteristic
4C2×#14523966C211yesmaximal, characteristic
5#16863292C211yesmaximal, characteristic
6#16863150C211yesmaximal, characteristic
7C2×#103×C67C211yesmaximal, characteristic
8C22×Dic134C2211yescharacteristic
9C22×C67⋊C8C2211yescharacteristic
10#14524041—22no—
11#14524041—22no—
12C2×C4⋊Dic67C2211yescharacteristic
13C2×Dic67⋊C4—21no—
14C2×Dic67⋊C4—21no—
15C2×Dic67⋊C4—21no—
16C2×C4×Dic67C2211yescharacteristic
17#14523968—22no—
18#14523967—22no—
19#14523966—22no—
20C67⋊#95—21no—
21C22×Q8×C67C2211yescharacteristic
22C22×C8×C67C2211yescharacteristic
23C2×C4⋊C4×C67C2211yescharacteristic
24#103×C67—22no—
25C2×Dic134—21no—
26C2×Dic134—21no—
27C2×Dic134—21no—
28C2×Dic134—21no—
29C2×Dic134—21no—
30C2×Dic134D412yes—
31C22×Dic67—21no—
32C22×Dic67—21no—
33C22×Dic67D411yescharacteristic
34C22×Dic67D411yescharacteristic
35C2×C67⋊C8—21no—
36C2×C67⋊C8—21no—
37C2×C67⋊C8—21no—
38C4⋊Dic67—21no—
39C4⋊Dic67D412yes—
40Dic67⋊C4—24no—
41Dic67⋊C4—24no—
42Dic67⋊C4—42no—
43C4×Dic67—21no—
44C4×Dic67—21no—
45C2×Q8×C67—21no—
46C2×Q8×C67—21no—
47C2×Q8×C67—21no—
48C2×Q8×C67—21no—
49C2×Q8×C67—21no—
50C2×Q8×C67D412yes—
51C22×C4×C67—21no—
52C22×C4×C67—21no—
53C22×C4×C67C2311yescharacteristic
54C2×C8×C67—21no—
55C2×C8×C67—21no—
56C2×C8×C67—21no—
57C4⋊C4×C67—21no—
58C4⋊C4×C67—21no—
59C23×C67C2×D411yescharacteristic
60Q8×C67—22no—
61Q8×C67—22no—
62Q8×C67—41no—
63Q8×C67—41no—
64C2×C4×C67—22no—
65C2×C4×C67—41no—
66C2×C4×C67—41no—
67C2×C4×C67—41no—
68C2×C4×C67—41no—
69C2×C4×C67—41no—
70C2×C4×C67C2×D411yescharacteristic
71C2×C4×C67C2×D411yescharacteristic
72C2×C4×C67C2×D411yescharacteristic
73C2×C4×C67#4111yescharacteristic
74C2×C4×C67#4111yescharacteristic
75C2×C4×C67#4111yescharacteristic
76C2×Dic67—21no—
77C2×Dic67—21no—
78C2×Dic67—21no—
79C2×Dic67—21no—
80C2×Dic67—22no—
81C2×Dic67—22no—
82C2×Dic67—41no—
83C2×Dic67—41no—
84C2×Dic67—41no—
85C2×Dic67—41no—
86C2×Dic67SD1612yes—
87C2×Dic67Q1612yes—
88Dic134—22no—
89Dic134—22no—
90Dic134—41no—
91Dic134—41no—
92C8×C67—41no—
93C67⋊C8—41no—
94C22×C67#12011yescharacteristic
95C22×C67#12111yescharacteristic
96C22×C67#12411yescharacteristic
97C22×C67C2×SD1611yescharacteristic
98C22×C67C2×Q1611yescharacteristic
99C22×C67#13711yescharacteristic
100C22×C67#13711yescharacteristic
101C4×C67—42no—
102C4×C67—81no—
103C4×C67#12111yescharacteristic
104C4×C67#12111yescharacteristic
105C4×C67#12311yescharacteristic
106C4×C67#12311yescharacteristic
107Dic67—22no—
108Dic67—22no—
109Dic67—42no—
110Dic67—42no—
111C2×C67#39411yescharacteristic
112C2×C67#44811yescharacteristic
113C2×C67#45111yescharacteristic
114C2×C67#46111yescharacteristic
115C2×C67#46211yescharacteristic
116C2×C67#48711yescharacteristic
117C2×C67#49111yescharacteristic
118#2043—671nomaximal, Sylow
119C67#204311yescharacteristic, complemented, Sylow
120C2×C4⋊Q8—671no—
121C2×#103—671no—
122C2×#103—671no—
123C2×#103—671no—
124C2×#103D6711yescharacteristic
125#389—671no—
126#336—671no—
127C22×Q8—671no—
128C22×Q8C2×D6711yescharacteristic
129C22×C8—671no—
130C22×C8C2×D6711yescharacteristic
131C4⋊Q8—1342no—
132C4⋊Q8—1342no—
133C2×C4⋊C4—671no—
134C2×C4⋊C4—1341no—
135C2×C4⋊C4—1341no—
136C2×C4⋊C4—1341no—
137C2×C4⋊C4C2×D6711yescharacteristic
138C2×C42—671no—
139#103—22no—
140#103—1342no—
141#103—1342no—
142#103—1342no—
143#95—1341no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—672no—
150C2×Q8—1341no—
151C2×Q8—1341no—
152C2×Q8—1341no—
153C2×Q8—1341no—
154C2×Q8—1341no—
155C2×Q8C22⋊D6712yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—671no—
159C22×C4—671no—
160C22×C4—1341no—
161C22×C4—1341no—
162C22×C4C22×D6711yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—1341no—
167C2×C8—1341no—
168C2×C8—1341no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—2682no—
172C4⋊C4—672no—
173C4⋊C4—1344no—
174C4⋊C4—1344no—
175C4⋊C4—1341no—
176C42—1341no—
177C42—1341no—
178C23D4×D6711yescharacteristic
179Q8—22no—
180Q8—22no—
181Q8—41no—
182Q8—41no—
183Q8—2681no—
184Q8—2681no—
185Q8—1342no—
186Q8—1342no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—2681no—
194C2×C4—2681no—
195C2×C4—2681no—
196C2×C4—2681no—
197C2×C4—672no—
198C2×C4—672no—
199C2×C4—1342no—
200C2×C4—1342no—
201C2×C4—1341no—
202C2×C4—1341no—
203C2×C4—1341no—
204C2×C4—1341no—
205C2×C4D4×D6711yescharacteristic
206C2×C4D4×D6711yescharacteristic
207C2×C4C2×C22⋊D6711yescharacteristic
208C2×C4#1176162711yescharacteristic
209C2×C4#1176162811yescharacteristic
210C2×C4#1176162811yescharacteristic
211C8—41no—
212C8—2681no—
213C22#1452401111yescharacteristic
214C22#1452401611yescharacteristic
215C22#1452402911yescharacteristic
216C22SD16×D6711yescharacteristic
217C22Q16×D6711yescharacteristic
218C22#1452407211yescharacteristic
219C22#1452407511yescharacteristic
220C4—42no—
221C4—81no—
222C4—2682no—
223C4—2682no—
224C4—1342no—
225C4—1342no—
226C4#1452401511yescharacteristic
227C4#1452401611yescharacteristic
228C4#1452402511yescharacteristic
229C4#1452402611yescharacteristic
230C2#1686331111yescharacteristic
231C2#1686353311yescharacteristic
232C2#1686355011yescharacteristic
233C2#1686360011yescharacteristic
234C2#1686360511yescharacteristic
235C2#1686374511yescharacteristic
236C2#1686376311yescharacteristic
subgroupG/H
centerC23D4×D67
commutatorC22×C4×C67C23
FrattiniC22×C4C22×D67
FittingC2×#103×C67C2
socleC23×C67C2×D4
hypercenterC2×#103D67
hypocenterC67#2043
LCSG ▷ C22×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
119. C67#2043semidirect